Theorem tgidm 15098

Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
tgidm	⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Proof of Theorem tgidm

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgvalex 13594	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
2		eltg3 15081	. . . . 5 ⊢ ((topGen‘𝐵) ∈ V → (𝑥 ∈ (topGen‘(topGen‘𝐵)) ↔ ∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦)))
3	1, 2	syl 14	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘(topGen‘𝐵)) ↔ ∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦)))
4		uniiun 4061	. . . . . . . . . 10 ⊢ ∪ 𝑦 = ∪ 𝑧 ∈ 𝑦 𝑧
5		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → 𝑦 ⊆ (topGen‘𝐵))
6	5	sselda 3248	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ (topGen‘𝐵))
7		eltg4i 15079	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (topGen‘𝐵) → 𝑧 = ∪ (𝐵 ∩ 𝒫 𝑧))
8	6, 7	syl 14	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) ∧ 𝑧 ∈ 𝑦) → 𝑧 = ∪ (𝐵 ∩ 𝒫 𝑧))
9	8	iuneq2dv 4028	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑧 ∈ 𝑦 𝑧 = ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧))
10	4, 9	eqtrid 2283	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 = ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧))
11		iuncom4 4014	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ 𝑦 ∪ (𝐵 ∩ 𝒫 𝑧) = ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧)
12	10, 11	eqtrdi 2287	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 = ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧))
13		inss1 3451	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
14	13	rgenw 2605	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
15		iunss 4048	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵 ↔ ∀𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵)
16	14, 15	mpbir 146	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵
17	16	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ⊆ (topGen‘𝐵) → ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵)
18		eltg3i 15080	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ⊆ 𝐵) → ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ∈ (topGen‘𝐵))
19	17, 18	sylan2 286	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ ∪ 𝑧 ∈ 𝑦 (𝐵 ∩ 𝒫 𝑧) ∈ (topGen‘𝐵))
20	12, 19	eqeltrd 2315	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → ∪ 𝑦 ∈ (topGen‘𝐵))
21		eleq1 2301	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 ∈ (topGen‘𝐵) ↔ ∪ 𝑦 ∈ (topGen‘𝐵)))
22	20, 21	syl5ibrcom 157	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ⊆ (topGen‘𝐵)) → (𝑥 = ∪ 𝑦 → 𝑥 ∈ (topGen‘𝐵)))
23	22	expimpd 363	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ((𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ (topGen‘𝐵)))
24	23	exlimdv 1872	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦(𝑦 ⊆ (topGen‘𝐵) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ (topGen‘𝐵)))
25	3, 24	sylbid 150	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘(topGen‘𝐵)) → 𝑥 ∈ (topGen‘𝐵)))
26	25	ssrdv 3254	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) ⊆ (topGen‘𝐵))
27		bastg 15085	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
28		tgss 15087	. . 3 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐵 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐵)))
29	1, 27, 28	syl2anc 415	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐵)))
30	26, 29	eqssd 3265	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3685  ∪ cuni 3930  ∪ ciun 4007  ‘cfv 5372  topGenctg 13585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-topgen 13591

This theorem is referenced by:  tgss3  15102  txbasval  15291